Skew-rank of an oriented graph and independence number of its underlying graph
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An oriented graph \(G^\sigma \) is a digraph which is obtained by orienting every edge of a simple graph G, where G is called the underlying graph of \(G^\sigma \). Let \(S(G^\sigma )\) denote the skew-adjacency matrix of \(G^\sigma \) and \(\alpha (G)\) be the independence number of G. The rank of \(S(G^\sigma )\) is called the skew-rank of \(G^\sigma \), denoted by \(sr(G^\sigma )\). Wong et al. studied the relation between the skew-rank of an oriented graph and the rank of its underlying graphs. Huang et al. recently studied the relationship between the skew-rank of an oriented graph and the independence number of its underlying graph, by giving some lower bounds for the sum, difference and quotient etc. They left over some questions for further studying the upper bounds of these parameters. In this paper, we extend this study by showing that \(sr(G^\sigma )+ 2\alpha (G) \le 2n\), where n is the order of G, and two classes of oriented graphs are given to show that the upper bound 2n can be achieved. Furthermore, we answer some open questions by obtaining sharp upper bounds for \(sr(G^\sigma ) + \alpha (G)\), \(sr(G^\sigma ) - \alpha (G)\) and \(sr(G^\sigma ) / \alpha (G)\).
KeywordsSkew-rank Oriented graph Oddly-oriented Independence number
Mathematics Subject Classification05C50
The authors are very grateful to the reviewers and editors for their detailed suggestions and comments which are very helpful to improve our paper.
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